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3. The following real numbers have decimal expansions as given below. In each casedecide whether they are rational or not. If they are rational, and of the formyou say about the prime factors of q?(1) 43.123456789(ii) 0.120120012000120000...(1) 43.123456789 |
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Answer» Answer: 2^n × 5^m form Step-by-step explanation: i ) Let X = p/q be a RATIONAL number , such that the prime FACTORISATION of q is of the form 2ⁿ × 5^m , where n and m are non - negative integers . Then x has a decimal expansion which terminates. ii ) The number which is non - terminating and non - repeating is called an irrational number. ******************************************* i ) x = 43.123456789 is a rational . x = 43123456789/( 1000000000 ) = 43123456789/( 10^9 ) = 43123456789/( 2 × 5 )^9 = 43123456789/( 2^9 × 5^9 ) Here , q = 2^9 × 5^9 ( 2ⁿ × 5^m form ) 43.123456789 is a terminating decimal. ii ) 0.120120012000120000.... is non - terminating and non - repeating decimal . THEREFORE , it is an irrational number. iii ) 43.123456789123456789.... is a non - terminating , repeating decimal. So it is a rational number. x = 43.123455789123456789....---( 1 ) 10^9 x = 43123456789.123456789....--(2 ) subtracting ( 1 ) from ( 2 ) , we get 10^9 x = 43123456746 x = 43123456746/10^9 x = 43123456746/( 2 × 5 )^9 x = 43123456746/( 2^9 × 5^9 ) Therefore , q = 2^n × 5^m form Given number is a rational . I hope this helps you |
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